Complex Numbers And Quadratic Equations question 436
Question: Let $ z_1,z_2 $ be two complex numbers such that $ z_1+z_2 $ and $ z_1z_2 $ both are real, then [RPET 1996]
Options:
A) $ z_1=-z_2 $
B) $ z_1={{\bar{z}}_2} $
C) $ z_1=-{{\bar{z}}_2} $
D) $ z_1=z_2 $
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ z_1=a+ib,z_2=c+id $ , then $ z_1+z_2 $ is real
Þ $ (a+c)+i(b+d) $ is real
Þ $ b+d=0 $
Þ $ d=-b $ …..(i) $ z_1z_2 $ is real
Þ $ (ad-bd)+i(ac+bc) $ is real
Þ $ ad+bc=0 $
Þ $ a(-b)+bc=0 $
Þ $ a=c $ \ $ z_1=a+ib=c-id={{\bar{z}}_2} $ $ (\because a=c $ and $ b=-d) $
BETA
